Taxes on revenue create strong incentives for vertical integration and consolidation (because there are fewer links in the chain to be taxed).

If one entity owns the farm, the food distribution, and the grocery store, they have one revenue transaction to be taxed. A small farmer selling their eggs to a distributor who sells them to the grocer who sells them to you is taxed three times on what amounts to same activity.

This is perhaps true of revenue tax. But for VAT You can deduct VAT of sales from VAT of purchases.

Indeed. Which is why VAT is a much better and more common structure of taxation than a tax on all revenue.

Most countries on earth--US being a notable exception--have a better version of this: the VAT.

It's a very elegant tax on paper, but significantly complicated from an accounting POV. It's one of those areas where there are huge gains to be had from digitizing financial records, and countries that have succeeded in this (like Mexico) create a very powerful revenue source.

Why tax a corporation's income at all when you can tax its shareholders income instead?

Well, the shareholders may live in a different area or country than the one in which the company operates. If a corporation is largely owned by American investors, but does its production largely in a developing nation (relying on their infrastructure to operate), then I think it's fair to say that both the developing country and the US should both get a slice of the pie: 1 for providing the infrastructure and labor market, and the other for providing the comforts of a developed country to shareholders.